A potential projection method for the nonlinear Schrödinger equation with low regularity potential
Abstract
We propose a potential projection (PP) method for the nonlinear Schrödinger equation (NLSE) with low-regularity and singular potentials by replacing the potential with its Fourier projection (truncated Fourier series). This gives rise to an NLSE with projected potential (NLSEwPP). We rigorously establish its well-posedness under low-regularity potentials, and its convergence to the original NLSE. The NLSEwPP is then directly discretized by the standard time-splitting Fourier pseudospectral (TS-FP) method with the mesh size chosen according to the potential projection parameter, resulting in the fully discrete TS-PP-FP method. Numerically, TS-PP-FP consistently achieves optimal temporal and spatial convergence rates with respect to the regularity of the exact solution under low-regularity and even singular potentials. Compared with the time-splitting Fourier spectral method, TS-PP-FP attains the same convergence rates at significantly lower computational cost and preserves the mass conservation and time symmetry (that are lost in the Fourier spectral discretization of the time-splitting method); compared with the time-splitting FP method, it is substantially more accurate for low-regularity potentials with the same computational cost.
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BibTeX
@misc{bao2026potential,
title = {{A potential projection method for the nonlinear Schrödinger equation with low regularity potential}},
author = {Weizhu Bao and Qingyuan Sun and Chushan Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19430},
eprint = {2610.19430},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19430},
}
RIS
TY - UNPB
TI - A potential projection method for the nonlinear Schrödinger equation with low regularity potential
AU - Weizhu Bao
AU - Qingyuan Sun
AU - Chushan Wang
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19430
UR - https://arxiv.org/abs/2610.19430
AB - We propose a potential projection (PP) method for the nonlinear Schrödinger equation (NLSE) with low-regularity and singular potentials by replacing the potential with its Fourier projection (truncated Fourier series). This gives rise to an NLSE with projected potential (NLSEwPP). We rigorously establish its well-posedness under low-regularity potentials, and its convergence to the original NLSE. The NLSEwPP is then directly discretized by the standard time-splitting Fourier pseudospectral (TS-FP) method with the mesh size chosen according to the potential projection parameter, resulting in the fully discrete TS-PP-FP method. Numerically, TS-PP-FP consistently achieves optimal temporal and spatial convergence rates with respect to the regularity of the exact solution under low-regularity and even singular potentials. Compared with the time-splitting Fourier spectral method, TS-PP-FP attains the same convergence rates at significantly lower computational cost and preserves the mass conservation and time symmetry (that are lost in the Fourier spectral discretization of the time-splitting method); compared with the time-splitting FP method, it is substantially more accurate for low-regularity potentials with the same computational cost.
ER -